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Evaluate lim(ntooo) (1^(3)+2^(3)+3^(3)+....

Evaluate `lim_(ntooo) (1^(3)+2^(3)+3^(3)+...+n^(3))/(sqrt(4n^(8)+1)).`

Text Solution

Verified by Experts

`underset(ntooo)lim(1^(3)+2^(3)+3^(3)+...+n^(3))/(sqrt(4n^(8)+1))=1/4underset(ntooo)lim(n^(2)(n+1)^(2))/(sqrt(4n^(8)+1))`
`=1/4underset(ntooo)lim(n^(2))/((1)/(n^(4))sqrt(4n^(8)+1))`
`=1/4underset(ntooo)lim(1+(1)/(n))^(2)/(sqrt(4+(1)/(n^(8))))=1/8`
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